Polytopes
Abstract
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Meshes.Polytope — Type
Polytope{K,M,CRS}We say that a geometry is a K-polytope when it is a collection of "flat" sides that constitute a K-dimensional subspace. They are called chain, polygon and polyhedron respectively for 1D (K=1), 2D (K=2) and 3D (K=3) subspaces. The parameter K is also known as the rank or parametric dimension of the polytope (https://en.wikipedia.org/wiki/Abstract_polytope).
The term polytope expresses a particular combinatorial structure. A polyhedron, for example, can be decomposed into faces. Each face can then be decomposed into edges, and edges into vertices. Some conventions act as a mapping between vertices and higher dimensional features (edges, faces, cells…), removing the need to store all features.
Additionally, the following property must hold in order for a geometry to be considered a polytope: the boundary of a (K+1)-polytope is a collection of K-polytopes, which may have (K-1)-polytopes in common. See https://en.wikipedia.org/wiki/Polytope.
Notes
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Type aliases are
Chain,Polygon,Polyhedron.
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Meshes.Chain — Type
Chain{M,CRS}A chain is a 1-polytope, i.e. a polytope with parametric dimension 1. See https://en.wikipedia.org/wiki/Polygonal_chain.
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Meshes.Polyhedron — Type
Polyhedron{M,CRS}A polyhedron is a 3-polytope, i.e. a polytope with parametric dimension 3.
See also Tetrahedron, Hexahedron and Pyramid.
Concrete
Ngon
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Meshes.Ngon — Type
Ngon(p₁, p₂, ..., pₙ)A N-gon is a polygon with N ≥ 3 vertices p₁, p₂, …, pₙ oriented counter-clockwise (CCW). In this case the number of vertices is fixed and known at compile time. Examples of N-gon are Triangle (N=3), Quadrangle (N=4), Pentagon (N=5), etc.
Notes
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Although the number of vertices
Nis known at compile time, we use abstract vectors to store the list of vertices. This design allows constructing N-gon from views of global vectors without expensive memory allocations. -
Type aliases are
Triangle,Quadrangle,Pentagon,Hexagon,Heptagon,Octagon,Nonagon,Decagon.
Triangle((0.,0.), (1.,0.), (0.,1.)) |> viz
PolyArea
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Meshes.PolyArea — Type
PolyArea(outer)
PolyArea([outer, inner₁, inner₂, ..., innerₖ])A polygonal area with outer ring, and optional inner rings inner₁, inner₂, …, innerₖ.
Rings can be a vector of Point or a vector of tuples with coordinates for convenience, in which case the first point should not be repeated at the end of the vector.
outer = [(0, 0), (1, 0), (1, 1), (0, 1)]
hole1 = [(0.2, 0.2), (0.2, 0.4), (0.4, 0.4), (0.4, 0.2)]
hole2 = [(0.6, 0.2), (0.6, 0.4), (0.8, 0.4), (0.8, 0.2)]
poly = PolyArea([outer, hole1, hole2]) |> viz
